Metric pairs and tuples in theory and applications

We present theoretical properties of the space of metric pairs equipped with the Gromov–Hausdorff distance. First, we establish the classical metric separability and the geometric geodesicity of this space. Second, we prove an Arzelà–Ascoli-type theorem for metric pairs. Third, extending a result by Cassorla, we show that the set of pairs consisting of a $2$-dimensional compact Riemannian manifold and a $2$-dimensional submanifold with boundary that can be isometrically embedded in $\mathbb{R}^3$ is dense in the space of compact metric pairs. Finally, to broaden the scope of potential applications, we describe scenarios where the Gromov–Hausdorff distance between metric pairs or tuples naturally arises.

💡 Research Summary

This paper develops a systematic theory of the Gromov–Hausdorff (GH) distance for metric pairs (X,A) and metric tuples (X,X_k,…,X_1), extending the classical GH framework that compares whole compact metric spaces. After recalling the classical Hausdorff and GH distances, the authors define a metric pair as a compact metric space together with a closed distinguished subset, and a tuple as a nested sequence of such subsets. The GH distance between two compact pairs is introduced as the infimum of Hausdorff distances over all admissible metrics on the disjoint union, and equivalently as one half of the minimal distortion of a “pair correspondence” that respects both the ambient spaces and the distinguished subsets. This reformulation mirrors the well‑known correspondence‑distortion characterization for ordinary GH distance and provides a concrete tool for analysis and computation.

The core geometric results are threefold. First, the space (GH₁,d_GH) of isometry classes of compact metric pairs is shown to be geodesic. By selecting an optimal correspondence R_opt and defining a linear interpolation metric d_γR(t) that blends the distances of the two spaces with weight t, the authors construct explicit geodesics between any two points, proving that the length of this curve equals the GH distance. Second, they establish separability: every compact pair can be approximated arbitrarily closely by pairs built from rational points and rational distances, yielding a countable dense subset. Third, they prove an Arzelà–Ascoli type theorem for relative maps between pairs. If a sequence of equi‑Lipschitz relative maps f_i:(X_i,A_i)→(Y_i,B_i) converges in the GH sense, then a subsequence converges to a continuous relative map f_∞:(X_∞,A_∞)→(Y_∞,B_∞) with the same Lipschitz bounds. This extends classical compactness of equicontinuous families to the setting where both domain and codomain carry distinguished subsets.

A significant application is a density theorem extending Cassorla’s result. Let M be the set of all compact length metric pairs and let S be the subset consisting of pairs where the ambient space is a 2‑dimensional compact Riemannian manifold that can be isometrically embedded in ℝ³ together with a 2‑dimensional submanifold with boundary. The authors prove that S is dense in M with respect to d_GH. In other words, any compact metric pair can be approximated arbitrarily well by a surface‑pair that admits an isometric embedding into three‑dimensional Euclidean space. The proof uses smooth approximations and triangulations to replace arbitrary metric structures by embedded surfaces while controlling the GH distance.

Finally, the paper surveys several contexts where GH distances between metric pairs or tuples arise naturally. These include hypernetwork and graph models, simplicial complexes, and spaces of persistence diagrams in topological data analysis. In each case the pair or tuple structure captures additional hierarchical or boundary information that ordinary GH distance on spaces would ignore, and the authors argue that their framework yields more refined stability results and a richer geometric language for applications in physics (e.g., spacetime models with distinguished hypersurfaces), network science, and computational geometry.

Overall, the work establishes fundamental metric‑geometric properties (geodesicity, separability, compactness of map families) for the space of metric pairs, proves a powerful density result for surface‑pairs, and points to a broad spectrum of potential applications, thereby laying a solid foundation for future research on GH distances in more structured settings.